I think the most simple solution is additional tanks plumbed in tandem. It reduces the amount of vapors being consumed from one tank to two tanks per burner. This reduced the freezing caused by excessive consumption of the vapors.
Still, warm water would help.

I'm changing my answer. Or coming around, maybe. The water doesn't scare me now as much as it just sounds like a horrific PITA. (Also, as someone with Reynaud's, dealing with water outside in the winter sounds near to impossible to me.)

I do like, and now understand, the idea of stringing many tanks together.

A glass-blowing friend, who lives an hour down the road (and previously used propane but has now switched to some sort of magical wood process that produces zero smoke), is encouraging me to go with one 100-lb tank. I previously stated I couldn't go this big, but it turns out it's still too small for the propane company to come and fill. Which is great, as I'm still on my own. It just leaves me to find a way to transport such an animal.

Perhaps people are suggesting water because when you move heated air there quite often a chance that a spark from any given source might blow into the tank area and cause an explosion. Also, you don't have to actually have the water exposed. You could dedicate some garden hoses and attach them to a sock that you set the tank in. Then you just have to circulate warmed water around the tanks and empty them when you're done. You could perhaps do this from and into a container that contacts the kiln to capture the waste heat. Note, you should always take precautions to not trap explosive gases, keep the tanks well ventilated.

This topic has been discussed inside out and upside down over the past few days around our house. There's only so far the two of us can take it, knowing so little about it all. I'm so very appreciative for all your knowledgeable help thus far.

This morning my husband and I are thinking we'll get two or three (or four) smaller tanks (50-lb? - whatever Marc says we need) and build a little doghouse for them - one that's very well ventilated. The little house will simply be a way to keep the snow off, so I'm not out there digging them out prior to a firing.

When it comes time to fire: Take a bucket of very warm water out there and add it to a little warming system, which consists of garden hose wrapped multiple times around each tank, a "keep the water warm for the livestock" unit in the bucket, and a pump circulating the water 'round.

Update: I spoke to Marc again. He told me, amongst other important tidbits: 1. Stop worrying about it and just try it. 2. Heating the exterior of the tanks only deals with the symptom, not the problem. Fix the problem by using more propane (more tanks or bigger tank[s]). 3. Stop asking questions on Internet forums. Ha! He's awesome.